[Paper Review] Implied Correlation for Pricing multi-FX options

Pavel V. Shevchenko|ArXiv.org|Apr 30, 2009

Monetary Policy and Economic Impact3 citations

TL;DR

This paper derives a formula for implied correlation between foreign exchange rates (FXRs) with different base currencies, enabling consistent pricing of multi-FX options—especially barrier and path-dependent derivatives—by aligning with market-quoted implied volatilities. The method extends the concept of implied correlation from 'currency triangles' to cross-currency FXR pairs, resolving inconsistencies in valuation when historical correlations are unstable or misaligned with vanilla option prices.

ABSTRACT

Option written on several foreign exchange rates (FXRs) depends on correlation between the rates. To evaluate the option, historical estimates for correlations can be used but usually they are not stable. More significantly, pricing of the option using these estimates is usually inconsistent to the traded vanilla contracts. To price options written on several FXRs with the same denominating currency, financial practitioners and traders often use implied correlations calculated from implied volatilities of FXRs that form "currency triangles". However, some options may have underlying FXRs with different denominating currencies. In this paper, we present the formula for the implied correlations between such FXRs. These can be used for valuation, for example, barrier option on two FXRs with different denominating currencies where one FXR determines how much the option is in or out of the money at maturity while another FXR is related to the barrier. Other relevant options are straightforward.

Motivation & Objective

• To address the inconsistency between historical correlation estimates and market prices of vanilla FX options in multi-currency derivatives.
• To extend the concept of implied correlation—commonly used in currency triangles—to FX rates with different denominating currencies.
• To provide a formula for implied correlation that supports accurate valuation of complex multi-FX options, such as barrier options.
• To resolve pricing discrepancies arising when using unstable historical correlations in multi-asset FX derivatives.

Proposed method

• Derives a closed-form expression for implied correlation between two FX rates that share a common quote currency but have different base currencies.
• Uses the relationship between implied volatilities of three FX rates forming a currency triangle to infer the implied correlation between non-traded FX pairs.
• Applies the no-arbitrage condition to express the implied correlation as a function of observed implied volatilities of the three FX rates.
• Validates the formula by ensuring consistency with the pricing of vanilla options and the absence of arbitrage in the FX market.
• Applies the derived correlation to price path-dependent options, such as barrier options, where one FX rate determines the option's payoff condition and another governs the barrier level.

Experimental results

Research questions

• RQ1 How can implied correlation be consistently calculated for FX rates with different denominating currencies, where traditional currency triangle methods do not apply?
• RQ2 What is the mathematical relationship between the implied volatilities of three FX rates forming a cross-currency triangle and the implied correlation between non-directly traded FX pairs?
• RQ3 How can the derived implied correlation formula improve the pricing of multi-FX options, especially barrier options with mixed-denomination underlying FX rates?
• RQ4 To what extent does using implied correlation instead of historical correlation reduce pricing errors in multi-asset FX derivatives?

Key findings

• The paper derives a closed-form formula for implied correlation between two FX rates with different base currencies, based on the implied volatilities of the three FX rates in a currency triangle.
• The derived implied correlation ensures consistency with the market prices of vanilla FX options, eliminating pricing discrepancies caused by unstable historical correlations.
• The method enables accurate valuation of complex derivatives such as barrier options where one FX rate determines the option's moneyness and another governs the barrier condition.
• The formula is validated through application to real-world market data, showing alignment with reported market quotes in Derivatives Week (2006).